Midterm 1
Midterm 2
Maxima/Minima
Lagrange Multipliers
100

Let u = < 1, 3, 1 > and v = <-3, 1, 0 >. Calculate v • u

0

100

Find the largest possible domain for f(x,y) = log(xy)

D = { (x,y) in R2 | (x>0 and y>0) or (x<0 and y<0) }

100

Find the critical point(s) of x2+y2-2x+4y+5

(1,-2)
100

Identify the function and the constraint in the following question: "Using multivariable calculus, find three positive numbers whose sum is 75 such that the sum of their squares is as small as possible."

Constraint: x + y + z = 75

Function: f(x,y,z) = x2 + y2 + z2

200

Let u = < 2t, -3, 9 > and v= < 0, - t2 , -3 >. Calculate all values of t, if they exist, such that u and v are orthogonal.

t = 3, t = -3

200

f(x,y) = 3x2y + ln(x) - cos(y). Find ∇ f(1,0)

<1, 3>

200

Conceptual check: How do we know if a certain critical point is the location of a maximum? What are the criteria?

If the determinant of the Hessian matrix (D in second derivative test) is greater than 0 AND the sign of fxx/fyy is negative

200

Solve the system for a point (x,y):

2x = λ

4y = λ

x + y = 1

( 2/3 , 1/3 )

300

Let u = < 1, 3, -2 >, and let v = < 1, -2, 1 >. Find the area of the parallelogram formed by u and v

√35

300

f(x,y,z) = cos(xy2z) - sin(x3y+z). What is fz(1,π,0)?

1

300

(0,0) is a critical point of f(x,y) = x3 + y3. Is it a minimum, maximum, or neither?

Inconclusive test

300

Find the minimum value of f(x,y) = x2+y2 on the hyperbola xy=1.

2

400

r(t) = 7t i + 0.5*sin(2t) j + 0.5*cos(2t)k. Calculate the velocity v(t) AND acceleration a(t)

v(t) = < 7, cos(2t), -sin(2t) >

a(t) = < 0, -2sin(2t), -2cos(2t) >

400

Find the tangent plane to f(x,y) = tan-1(yex) at the point (0,1)

L(x,y) = (π/4) + (x/2) + (1/2)(y - 1)

400

Give that the gradient of f is < 2y + 3, 2x + 4 >, find the (single) critical point and determine if it is min/max/saddle/inconclusive

(-2,-3/2) is a saddle point

400

Use the method of Lagrange multipliers to find the minimum value of x2+4y2-2x+8y subject to the constraint x + 2y = 7

minimum = 27 at (5,1)

500

Does the line < 0, 1, 5 > + < 4, -2, 0 >t lie on the plane x + 2y - z + 3 = 0?

Yes

500

f(x,y) = 3x2+y ;  x(s,t) = 2s + t ;  y(s,t) = s - t ;  Find fs evaluated at (s,t) = (1, -2)

1

500

Calculate the critical points of 4x2+9y2+8x-36y+24 and determine if they are min/max/saddle/inconclusive

(-1,2) is a minimum

500

Find the max AND min of xyz subject to the constraint x2+2y2+3z2=6

max: 2/√3

min: -2/√3

M
e
n
u